Mathematics

Problem

Let

\mathbf{a}

\mathbf{b}

\mathbf{c}

be non-zero vectors and

|\mathbf{a}| = 1

and

\mathbf{r}

is a non-zero vector such that

\mathbf{r} \times \mathbf{a} = \mathbf{b}

and

\mathbf{r} \cdot \mathbf{c} = 1

, then a.

\mathbf{a} \perp \mathbf{b}

\mathbf{r} \perp \mathbf{b}

\mathbf{r} \cdot \mathbf{a} = \frac{1 - [\mathbf{a} \mathbf{b} \mathbf{c}]}{\mathbf{a} \cdot \mathbf{b}}

[\mathbf{r} \mathbf{a} \mathbf{b}] = 0